Geometry Chapter 9 Notes: Polygons and Quadrilaterals

Parallelograms

  • Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
  • Properties:
    • Opposite sides are parallel.
    • Opposite sides are congruent (equal in length).
    • Opposite angles are congruent (equal in measure).
    • Consecutive angles are supplementary (their measures add up to 180^").
    • Diagonals bisect each other (they cut each other in half).
    • A diagonal divides a parallelogram into two congruent triangles.

Proving Parallelograms

  • Method 1: Show that the diagonals bisect each other by demonstrating that the midpoints of the diagonals are the same.
  • Method 2: Show that both pairs of opposite sides are parallel by showing they have equal slopes.
  • Method 3: Show that both pairs of opposite sides are equal in length by using the distance formula.
  • Method 4: Show that one pair of sides is both parallel and congruent.

Rectangles

  • Definition: A rectangle is a parallelogram that contains one right angle.
  • Key Property: The diagonals of a rectangle are congruent.
  • Inherited Properties: All properties that apply to parallelograms also apply to rectangles.

Proving Rectangles

  • Method 1: Show that the diagonals are congruent.
  • Method 2: Show that the parallelogram has a right angle (e.g., by showing adjacent sides have opposite reciprocal slopes).

Rhombus

  • Definition: A rhombus is a parallelogram with all four sides congruent.
  • Key Properties:
    • The diagonals are perpendicular to each other.
    • The diagonals bisect the angles of the rhombus.

Proving Rhombuses

  • Method 1: Prove that the diagonals are perpendicular bisectors of each other.
  • Method 2: Prove that the quadrilateral is a parallelogram with a pair of adjacent sides congruent.
  • Method 3: Prove that all four sides are equal in length.

Squares

  • Definition: A square is a quadrilateral that is both a rhombus and a rectangle.
  • Properties: A square has all the properties of a parallelogram, a rectangle, and a rhombus.

Proving Squares

  • Method: Show that all four sides are congruent and that the quadrilateral has a right angle.

Trapezoids

  • Definition: A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel.
    • Parallel sides are called bases.
  • Isosceles Trapezoid:
    • Definition: A trapezoid in which the non-parallel sides (legs) are congruent.
    • Properties:
      • The base angles are congruent.
      • The diagonals are congruent.
  • Median (Midsegment) of a Trapezoid:
    • Definition: The segment connecting the midpoints of the non-parallel sides.
    • Property: The length of the median is half the sum of the lengths of the parallel sides. If the lengths of the parallel sides are b1andandb2,thenthelengthofthemedian, then the length of the medianmisgivenby:is given by:m = \frac{1}{2}(b1 + b2)$$

Proving Trapezoids

  • Method: Show that one pair of sides is parallel (same slope) and the other pair of sides is not parallel (different slopes).

Proving Isosceles Trapezoids

  • First, prove that the quadrilateral is a trapezoid.
  • Then, use one of the following methods:
    • Method 1: Prove that the diagonals are congruent (using the distance formula).
    • Method 2: Prove that the non-parallel sides are congruent (using the distance formula).